Noneuclidean Tessellations and Their Relation to Regge Trajectories

The coefficients in the confluent hypergeometric equation specifythe Regge trajectories and the degeneracy of the angular momentum states. Boundstates are associated with real angular momenta while resonances arecharacterized by complex angular momenta. With a centrifugal potential, thehalf-plane is tessellated by crescents. The addition of an electrostaticpotential converts it into a hydrogen atom, and the crescents into triangleswhich may have complex conjugate angles; the angle through which a rotationtakes place is accompanied by a stretching. Rather than studying the propertiesof the wave functions themselves, we study their symmetry groups. A complexangle indicates that the group contains loxodromic elements. Since the domainof such groups is not the disc, hyperbolic plane geometry cannot be used.Rather, the theory of the isometric circle is adapted since it treats allgroups symmetrically. The pairing of circles and their inverses is likened topairing particles with their antiparticles which then go on to produce nestedcircles, or a proliferation of particles. A corollary to Laguerre’stheorem, which states that the euclidean angle is represented by a pureimaginary projective invariant, represents the imaginary angle in the form of areal projective invariant.